Analysis of Crosscutting in Early Software Development Phases Based on Traceability - Transactions on Aspect-Oriented Software Development - page 82

Each of these mappings can be described with a dependency matrix. We describe
how to combine two consecutive dependency matrices in an operation we call
cascading. Cascading is an operation on two dependency matrices resulting in a new
dependency matrix, which represents the dependency relation between source
elements of the first matrix and target elements of the second matrix.

Fig. 4. Cascading of consecutive levels

For cascading, it is essential to define the transitivity of dependency relations.
Transitivity is defined as follows. Assume we have a source, an intermediate level and
a target, as shown in Fig. 4. There is a dependency relation between an element in the
source and an element in the target if there is some element at the intermediate level
that has a dependency relation with this source element and a dependency relation
with this target element. In other words, the transitivity dependency relation f for
source s, intermediate level u and target t, where card(u) is the number of elements in
u, is defined as:
∃ k ∊ (1..card(u)): ( s[i] f u[k] )

( u[k] f t[m] ) ⇒ ( s[i] f t[m] ).
We can also formalize this relation in terms of the dependency matrices. Assume
we have three dependency matrices m1 :: s x u and m2 :: u x t and m3 :: s x t, where
s is the source, u is some intermediate level, card(u) is the cardinality of u, and t is the
target. The cascaded dependency matrix m3 is computed from matrices m1 and m2 as
follows: m3 = m1 x m2
Then , transitivity of the dependency relation is defined as follows:

∧

k ∊ (1..card(u)): m1[i,k] m2[k,m] ⇒ m3[i,m].

In terms of linear algebra, the dependency matrix is a relationship between two
given domains, source and target (see Section 0). Accordingly, the cascading
operation can be generalized as a composition of relationships as follows. Let Dom K ,
k = 1..n , be n domains, and let f i be the relationship between domains Dom i and
Dom i+1 , 1≤i<n , denoted as

∃ ∧

f

Dom . Let Source and Target be the domains
Dom 1 and Dom n , respectively. Consequently, we have the following relation-
ship between the domains:

i

⎯ ⎯→ Dom

i i + 1

f f f f

1 … .
As a result, the dependency relationship between the Source and the Target is defined
as

2 3 n − 1

S ource ⎯ ⎯→ Dom ⎯ ⎯→ Dom ⎯ ⎯→ Dom ⎯ ⎯→ T arget

2 3 n − 1

. In this way, the dependency matrix between a source and
target is obtained through matrix multiplication of the dependency matrices
representing each f i, 1≤i<n .
As an example, we explain the cascading of two dependency matrices: one for
concerns x requirements and one for requirements x modules. The two dependency

DM ≡ f f … f

n − 1 n − 2 1